6.003 Homework 12
Due at the beginning of recitation on
Wednesday, December 2, 2009
.
Problems
1. Sampling CT sinusoids
Consider 3 CT signals:
x
1
(
t
) = cos(3000
t
)
,
x
2
(
t
) = cos(4000
t
)
,
and
x
3
(
t
) = cos(5000
t
)
.
Each of these is sampled as follows
x
1
[
n
] =
x
1
(
nT
)
,
x
2
[
n
] =
x
2
(
nT
)
,
and
x
3
[
n
] =
x
3
(
nT
)
,
where
T
= 0
.
001
.
Which of the resulting DT signals has the highest DT frequency?
Which has the lowest DT frequency?
2. Sampling with alternating impulses
A CT signal
x
c
(
t
)
is converted to a DT signal
x
d
[
n
]
as follows:
x
d
[
n
] =
x
c
(
nT
)
n
even
−
x
c
(
nT
)
n
odd
a.
Assume that the Fourier transform of
x
c
(
t
)
is
X
c
(
jω
)
shown below.
ω
X
c
(
jω
)
W
1
Determine the DT Fourier transform
X
d
(
e
j
Ω
)
of
x
d
[
n
]
.
b.
Assume that
x
c
(
t
)
is bandlimited to
−
W
≤
ω
≤
W
. Determine the maximum value
of
W
for which the original signal
x
c
(
t
)
can be reconstructed from the samples
x
d
[
n
]
.
c.
Make a diagram of a system to reconstruct
x
c
(
t
)
from
x
d
[
n
]
.
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6.003 Homework 12 / Fall 2009
2
3. Boxcar sampling
A digital camera focuses light from the environment onto an imaging chip that converts
the incident image into a discrete representation composed of pixels. Each pixel represents
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 Spring '11
 DennisM.Freeman
 Digital Signal Processing, Trigraph, Continuous signal, Nyquist–Shannon sampling theorem

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